Last lecture Passive Stereo Spacetime Stereo Multiple View Stereo

Today Structure from Motion: Given pixel correspondences, how to compute 3D structure and camera motion? Slides stolen from Prof Yungyu Chuang

Epipolar geometry & fundamental matrix

The epipolar geometry What if only C,C’,x are known?

The epipolar geometry C,C’,x,x’ and X are coplanar epipolar geometry demo

The epipolar geometry All points on  project on l and l’

The epipolar geometry Family of planes  and lines l and l’ intersect at e and e’

The epipolar geometry epipolar plane = plane containing baseline epipolar line = intersection of epipolar plane with image epipolar pole = intersection of baseline with image plane = projection of projection center in other image epipolar geometry demo

The fundamental matrix F C C’ T=C’-C R p p’ The equation of the epipolar plane through X is

The fundamental matrix F essential matrix

The fundamental matrix F C C’ T=C’-C R p p’

The fundamental matrix F Let M and M’ be the intrinsic matrices, then fundamental matrix

The fundamental matrix F C C’ T=C’-C R p p’

The fundamental matrix F The fundamental matrix is the algebraic representation of epipolar geometry The fundamental matrix satisfies the condition that for any pair of corresponding points x↔x’ in the two images

F is the unique 3x3 rank 2 matrix that satisfies x’ T Fx=0 for all x ↔ x’ 1.Transpose: if F is fundamental matrix for (P,P’), then F T is fundamental matrix for (P’,P) 2.Epipolar lines: l’=Fx & l=F T x’ 3.Epipoles: on all epipolar lines, thus e’ T Fx=0,  x  e’ T F=0, similarly Fe=0 4.F has 7 d.o.f., i.e. 3x3-1(homogeneous)-1(rank2) 5.F maps from a point x to a line l’=Fx (not invertible) The fundamental matrix F

It can be used for –Simplifies matching –Allows to detect wrong matches

Estimation of F — 8-point algorithm The fundamental matrix F is defined by for any pair of matches x and x’ in two images. Let x=(u,v,1) T and x’=(u’,v’,1) T, each match gives a linear equation

8-point algorithm In reality, instead of solving, we seek f to minimize, least eigenvector of.

8-point algorithm To enforce that F is of rank 2, F is replaced by F’ that minimizes subject to. It is achieved by SVD. Let, where, let then is the solution.

8-point algorithm % Build the constraint matrix A = [x2(1,:)‘.*x1(1,:)' x2(1,:)'.*x1(2,:)' x2(1,:)'... x2(2,:)'.*x1(1,:)' x2(2,:)'.*x1(2,:)' x2(2,:)'... x1(1,:)' x1(2,:)' ones(npts,1) ]; [U,D,V] = svd(A); % Extract fundamental matrix from the column of V % corresponding to the smallest singular value. F = reshape(V(:,9),3,3)'; % Enforce rank2 constraint [U,D,V] = svd(F); F = U*diag([D(1,1) D(2,2) 0])*V';

8-point algorithm Pros: it is linear, easy to implement and fast Cons: susceptible to noise

Problem with 8-point algorithm ~10000 ~100 1 ! Orders of magnitude difference between column of data matrix  least-squares yields poor results

Normalized 8-point algorithm (0,0) (700,500) (700,0) (0,500) (1,-1) (0,0) (1,1)(-1,1) (-1,-1) normalized least squares yields good results Transform image to ~[-1,1]x[-1,1]

Normalized 8-point algorithm 1.Transform input by, 2.Call 8-point on to obtain 3.

Normalized 8-point algorithm A = [x2(1,:)‘.*x1(1,:)' x2(1,:)'.*x1(2,:)' x2(1,:)'... x2(2,:)'.*x1(1,:)' x2(2,:)'.*x1(2,:)' x2(2,:)'... x1(1,:)' x1(2,:)' ones(npts,1) ]; [U,D,V] = svd(A); F = reshape(V(:,9),3,3)'; [U,D,V] = svd(F); F = U*diag([D(1,1) D(2,2) 0])*V'; % Denormalise F = T2'*F*T1; [x1, T1] = normalise2dpts(x1); [x2, T2] = normalise2dpts(x2);

Normalization function [newpts, T] = normalise2dpts(pts) c = mean(pts(1:2,:)')'; % Centroid newp(1,:) = pts(1,:)-c(1); % Shift origin to centroid. newp(2,:) = pts(2,:)-c(2); meandist = mean(sqrt(newp(1,:).^2 + newp(2,:).^2)); scale = sqrt(2)/meandist; T = [scale 0 -scale*c(1) 0 scale -scale*c(2) ]; newpts = T*pts;

RANSAC repeat select minimal sample (8 matches) compute solution(s) for F determine inliers until  (#inliers,#samples)>95% or too many times compute F based on all inliers

Results (ground truth)

Results (8-point algorithm)

Results (normalized 8-point algorithm)

From F to R, T If we know camera parameters Hartley and Zisserman, Multiple View Geometry, 2 nd edition, pp 259

Application: View morphing

Main trick Prewarp with a homography to rectify images So that the two views are parallel Because linear interpolation works when views are parallel

Problem with morphing Without rectification

prewarp morph homographies input output

Video demo

Richard SzeliskiCSE 576 (Spring 2005): Computer Vision 40 Triangulation Problem: Given some points in correspondence across two or more images (taken from calibrated cameras), {(u j,v j )}, compute the 3D location X

Richard SzeliskiCSE 576 (Spring 2005): Computer Vision 41 Triangulation Method I: intersect viewing rays in 3D, minimize: X is the unknown 3D point C j is the optical center of camera j V j is the viewing ray for pixel (u j,v j ) s j is unknown distance along V j Advantage: geometrically intuitive CjCj VjVj X

Richard SzeliskiCSE 576 (Spring 2005): Computer Vision 42 Triangulation Method II: solve linear equations in X advantage: very simple Method III: non-linear minimization advantage: most accurate (image plane error)

Structure from motion

structure from motion: automatic recovery of camera motion and scene structure from two or more images. It is a self calibration technique and called automatic camera tracking or matchmoving. Unknowncameraviewpoints

Applications For computer vision, multiple-view shape reconstruction, novel view synthesis and autonomous vehicle navigation. For film production, seamless insertion of CGI into live-action backgrounds

Structure from motion 2D feature tracking 3D estimation optimization (bundle adjust) geometry fitting SFM pipeline

Structure from motion Step 1: Track Features Detect good features, Shi & Tomasi, SIFT Find correspondences between frames –Lucas & Kanade-style motion estimation –window-based correlation –SIFT matching

Structure from Motion Step 2: Estimate Motion and Structure Simplified projection model, e.g., [Tomasi 92] 2 or 3 views at a time [Hartley 00]

Structure from Motion Step 3: Refine estimates “Bundle adjustment” in photogrammetry Other iterative methods

Structure from Motion Step 4: Recover surfaces (image-based triangulation, silhouettes, stereo…) Good mesh

Example : Photo Tourism

Factorization methods

Problem statement

Other projection models

SFM under orthographic projection 2D image point orthographic projection matrix 3D scene point image offset Trick Choose scene origin to be centroid of 3D points Choose image origins to be centroid of 2D points Allows us to drop the camera translation:

factorization (Tomasi & Kanade) projection of n features in one image: projection of n features in m images W measurement M motion S shape Key Observation: rank(W) <= 3

Factorization Technique –W is at most rank 3 (assuming no noise) –We can use singular value decomposition to factor W: Factorization –S ’ differs from S by a linear transformation A: –Solve for A by enforcing metric constraints on M known solve for

Metric constraints Orthographic Camera Rows of  are orthonormal: Enforcing “Metric” Constraints Compute A such that rows of M have these properties Trick (not in original Tomasi/Kanade paper, but in followup work) Constraints are linear in AA T : Solve for G first by writing equations for every  i in M Then G = AA T by SVD

Results

Extensions to factorization methods Paraperspective [Poelman & Kanade, PAMI 97] Sequential Factorization [Morita & Kanade, PAMI 97] Factorization under perspective [Christy & Horaud, PAMI 96] [Sturm & Triggs, ECCV 96] Factorization with Uncertainty [Anandan & Irani, IJCV 2002]

Bundle adjustment

Richard SzeliskiCSE 576 (Spring 2005): Computer Vision 62 Structure from motion How many points do we need to match? 2 frames: ( R,t ): 5 dof + 3n point locations  4n point measurements  n  5 k frames: 6(k–1)-1 + 3n  2kn always want to use many more

Richard SzeliskiCSE 576 (Spring 2005): Computer Vision 63 Bundle Adjustment What makes this non-linear minimization hard? many more parameters: potentially slow poorer conditioning (high correlation) potentially lots of outliers

Richard SzeliskiCSE 576 (Spring 2005): Computer Vision 64 Lots of parameters: sparsity Only a few entries in Jacobian are non-zero

Richard SzeliskiCSE 576 (Spring 2005): Computer Vision 65 Robust error models Outlier rejection use robust penalty applied to each set of joint measurements for extremely bad data, use random sampling [RANSAC, Fischler & Bolles, CACM’81]

Richard SzeliskiCSE 576 (Spring 2005): Computer Vision 66 Correspondences Can refine feature matching after a structure and motion estimate has been produced decide which ones obey the epipolar geometry decide which ones are geometrically consistent (optional) iterate between correspondences and SfM estimates using MCMC [Dellaert et al., Machine Learning 2003]Dellaert et al., Machine Learning 2003

Richard SzeliskiCSE 576 (Spring 2005): Computer Vision 67 Structure from motion: limitations Very difficult to reliably estimate metric structure and motion unless: large (x or y) rotation or large field of view and depth variation Camera calibration important for Euclidean reconstructions Need good feature tracker Lens distortion

Issues in SFM Track lifetime Nonlinear lens distortion Prior knowledge and scene constraints Multiple motions

Track lifetime every 50th frame of a 800-frame sequence

Track lifetime lifetime of 3192 tracks from the previous sequence

Track lifetime track length histogram

Nonlinear lens distortion

effect of lens distortion

Prior knowledge and scene constraints add a constraint that several lines are parallel

Prior knowledge and scene constraints add a constraint that it is a turntable sequence

Applications of Structure from Motion

Jurassic park

PhotoSynth